行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}2 & 1 & 1 & {-}2 & 0 \\ 1 & {-}2 & 2 & 2 & 1 \\ 0 & {-}1 & {-}2 & 1 & {-}2 \\ 0 & {-}1 & 2 & 0 & 2 \\ {-}1 & {-}2 & 2 & 1 & {-}1\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 & 0 \\ 1 & {-}2 & 2 & 2 & 1 \\ 0 & {-}1 & {-}2 & 1 & {-}2 \\ 0 & {-}1 & 2 & 0 & 2 \\ {-}1 & {-}2 & 2 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第5列で展開する : [0, 1, -2, 2, -1]

\( \qquad\quad = (-1)^{5-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 0 & {-}1 & {-}2 & 1 \\ 0 & {-}1 & 2 & 0 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| + (-1)^{5-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & 2 & 0 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| + (-1)^{5-4}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & {-}2 & 1 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| + (-1)^{5-5}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & {-}2 & 1 \\ 0 & {-}1 & 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{5-2}\times1\times1 + (-1)^{5-3}\times(-2)\times3 + (-1)^{5-4}\times2\times(-25) + (-1)^{5-5}\times(-1)\times(-17) \)

\( \qquad\quad = 60 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 0 & {-}1 & {-}2 & 1 \\ 0 & {-}1 & 2 & 0 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 1, 0, 1]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & {-}1 & {-}2 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-4}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times4 + (-1)^{4-2}\times1\times1 + (-1)^{4-4}\times1\times(-8) \)

\( \qquad\quad = 1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & {-}1 & {-}2 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times(-1) + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times2\times0 \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| = 0\times(-1) - (-1)\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times(-3) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times(-1) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times0 + (-1)^{3-2}\times(-2)\times(-2) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| = 0\times(-1) - (-1)\times0 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & 2 & 0 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, 0, 1]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-4}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times4 + (-1)^{4-2}\times2\times1 + (-1)^{4-4}\times1\times(-7) \)

\( \qquad\quad = 3 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times(-1) + (-1)^{3-2}\times2\times(-4) + (-1)^{3-3}\times2\times(-1) \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| = 1\times(-2) - (-2)\times(-1) = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times(-3) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times(-1) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times(-2) + (-1)^{3-3}\times2\times(-5) \)

\( \qquad\quad = -7 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times1 = -5 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & {-}2 & 1 \\ {-}1 & {-}2 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, 1, 1]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & {-}2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| + (-1)^{4-4}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times(-12) + (-1)^{4-2}\times2\times(-11) + (-1)^{4-3}\times1\times(-8) + (-1)^{4-4}\times1\times13 \)

\( \qquad\quad = -25 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & {-}2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times(-1) + (-1)^{3-2}\times(-2)\times(-4) + (-1)^{3-3}\times2\times(-1) \)

\( \qquad\quad = -12 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| = 1\times(-2) - (-2)\times(-1) = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times(-2)\times(-3) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = -11 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ {-}1 & {-}2\end{array}\,\right| = 0\times(-2) - (-1)\times(-1) = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times(-1) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ {-}1 & {-}2 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-4) + (-1)^{3-2}\times2\times(-3) + (-1)^{3-3}\times2\times(-5) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ {-}1 & {-}2\end{array}\,\right| = 1\times(-2) - (-2)\times(-1) = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ {-}1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times(-1) = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times1 = -5 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, -2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times(-2) + (-1)^{3-3}\times(-2)\times(-5) \)

\( \qquad\quad = 13 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times1 = -5 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 1 & 1 & {-}2 \\ 1 & {-}2 & 2 & 2 \\ 0 & {-}1 & {-}2 & 1 \\ 0 & {-}1 & 2 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, 1, 0]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| + (-1)^{4-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times(-4) + (-1)^{4-2}\times2\times(-8) + (-1)^{4-3}\times1\times(-7) \)

\( \qquad\quad = -17 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 2 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-2)\times(-1) + (-1)^{3-3}\times2\times(-1) \)

\( \qquad\quad = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| = 0\times(-1) - (-1)\times0 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 0 & {-}1 & {-}2 \\ 0 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times0 + (-1)^{3-2}\times(-2)\times(-2) + (-1)^{3-3}\times2\times(-2) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & {-}1 \\ 0 & {-}1\end{array}\,\right| = 0\times(-1) - (-1)\times0 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 1 & 1 \\ 1 & {-}2 & 2 \\ 0 & {-}1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, 2, 2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times(-1) + (-1)^{3-2}\times2\times(-2) + (-1)^{3-3}\times2\times(-5) \)

\( \qquad\quad = -7 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & {-}1\end{array}\,\right| = 1\times(-1) - (-2)\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 0 & {-}1\end{array}\,\right| = 2\times(-1) - 1\times0 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 1 \\ 1 & {-}2\end{array}\,\right| = 2\times(-2) - 1\times1 = -5 \)